In recent years, several authors studied the linear (additive) maps that behave like homomorphisms, derivations or right (left) centalizers when acting on special products (for instance, see [1]}, [2]}, [3]}, [4]}, [5]} and the references therein). An algebra \( \mathcal {A} \)  is called zero product determined if for every linear space \(\mathcal {X}\)  and every bilinear map \(\phi :\mathcal {A}\times \mathcal {A}\rightarrow \mathcal {X}\)  the following holds: If \(\phi (a,b)=0\)  whenever \(ab=0\) , then there exists a linear map \(T : \mathcal {A}^{2}\rightarrow \mathcal {X}\)  such that \(\phi (a,b)=T(ab)\)  for each \(a,b\in \mathcal {A}\) . If \(\mathcal {A}\)  has unity 1, then \(\mathcal {A}\)  is zero product determined if and only if for every linear space \(\mathcal {X}\)  and every bilinear map \(\phi :\mathcal {A}\times \mathcal {A}\rightarrow \mathcal {X}\) , the following holds: If \(\phi (a,b)=0\)  whenever \(ab=0\) , then \(\phi (a,b)=\phi (ab,1)\)  for each \(a,b\in \mathcal {A}\) . Also in this case \(\phi (a,1)=\phi (1,a)\)  for all \(a\in \mathcal {A}\) . The question of characterizing linear maps through zero products, Jordan products, etc. on algebras sometimes can be effectively solved by considering bilinear maps that preserve certain zero product properties (for instance, see [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}). Motivated by these works, Brešar et al. [15]} introduced the concept of zero product (Jordan product) determined algebras, which can be used to study linear maps preserving zero products (Jordan products) and derivable (Jordan derivable) maps at zero point.
